Solving Indefinite Integrals Involving Square Roots

Calculus often involves solving integrals that may include complex functions, such as square roots of trigonometric expressions. One particular challenge is the integral of (int sqrt{1 - sin x} , dx). This article will guide you through the detailed steps to solve such integrals, utilizing trigonometric identities and integration techniques.

Understanding the Problem

The integral in question is (int sqrt{1 - sin x} , dx). At first glance, it might seem daunting; however, by breaking down the expression, it becomes more manageable.

Step-by-Step Solution

Step 1: Rewrite the Sine Function

The expression (1 - sin x) can be rewritten using trigonometric identities. Notice that (sin^2 frac{x}{2} cos^2 frac{x}{2} 1), and this leads us to:

(sqrt{1 - sin x} sqrt{sin^2 frac{x}{2} cos^2 frac{x}{2} - 2 sin frac{x}{2} cos frac{x}{2}})

This can be simplified further:

(sqrt{1 - sin x} sqrt{left(sin frac{x}{2} - cos frac{x}{2}right)^2})

Since the square root function and the square cancel each other out:

(sqrt{left(sin frac{x}{2} - cos frac{x}{2}right)^2} left| sin frac{x}{2} - cos frac{x}{2} right|)

For the indefinite integral, we can consider both positive and negative solutions, but for simplicity, let's assume (sin frac{x}{2} geq cos frac{x}{2}). Thus, the absolute value can be removed.

(int sqrt{1 - sin x} , dx int left( sin frac{x}{2} - cos frac{x}{2} right) , dx)

Step 2: Integration

This integral can now be split into two simpler integrals:

(int left( sin frac{x}{2} - cos frac{x}{2} right) , dx int sin frac{x}{2} , dx - int cos frac{x}{2} , dx)

Using the standard integration techniques, we get:

(int sin frac{x}{2} , dx -2cos frac{x}{2} C_1)

(int cos frac{x}{2} , dx 2sin frac{x}{2} C_2)

Combining these results, we have:

(int left( sin frac{x}{2} - cos frac{x}{2} right) , dx -2cos frac{x}{2} - 2sin frac{x}{2} C)

where (C) is the constant of integration.

Final Result

The indefinite integral of (sqrt{1 - sin x}) is:

(int sqrt{1 - sin x} , dx -2left(cos frac{x}{2} sin frac{x}{2}right) C)

Conclusion

By using trigonometric identities and careful integration techniques, we can solve what initially appeared to be a complex integral. This example demonstrates the power of breaking down problems into simpler, more manageable parts.

Further Reading

For more challenging and interesting integrals, refer to:

Continuous Learning in Calculus Advanced Techniques in Integration

Keywords

indefinite integrals, square root integrals, trigonometric integrals